泛函分析 2012-2013春季学期 2010级数理科学

任课老师

基本信息

学分:3.0

学时:48

时间:单周:星期三10:00-11:40&星期五10:00-11:40;双周:星期五10:00-11:40

地点: 中院402

课程代码:MA345

教学大纲

Tyaglov Course Functional Analysis.rtf

AIMS & OBJECTIVES

This course provides an introduction to the linear functional analysis and some ele-ments of nonlinear functional analysis. It includes the theory of metric and normed spaces (as much as it necessary), Hahn–Banah theory, theory of compact and linear operators, Fredholm theory, and spectral theory of linear operators as well as derivatives of nonlinear operators. The course contains some example of applications of functional analysis to PDE, in particular to elasticity theory and fluid mechanics.

TEACHING METHOD

Lectures and assignments.

RELATION TO OTHER COURSES

It closely related to Fourier, complex, harmonic, and real analysis courses as well as to partial differential equations and mathematical physics.

PREREQUISITES

Real and complex analysis including basic knowledge of metric spaces, PDE.

COURSE OUTLINE

1.Introduction to metric spaces

1.1 Definitions: metric space, energy spaces, examples
1.2 Complete, compact and separable metric spaces

2.Linear, Normed and Banach spaces

2.1 Linear spaces
2.2 Normed spaces
2.3 Inner product
2.4 Banach spaces
2.5 Hilbert spaces
2.6 Lebesgue and Sobolev spaces

3.Linear Operators

3.1 Linear operators. Continuity and boundedness
3.2 Spaces of linear operators
3.3 Inverse operators

4.Dual spaces and operators

4.1 Hahn–Banach theorem and its consequences
4.2 Dual spaces and operators
4.3 Projections and complementary subspaces
4.4 Weak and weak-* convergence

5.Linear operators on Hilbert spaces

5.1 The Adjoint of an operator
5.2 Normal, Self-adjoint and Unitary operators

6.Compact sets and completely continuous operators

6.1 Compact sets in normed spaces
6.2 Linear completely continuous operators

7.Fredholm’s theory

8.Introduction to spectral theory of linear operators

8.1 Eigenvalues and eigenvectors of linear operators
8.2 The resolvent set and the spectrum of a linear operator
8.3 Spectral decomposition of self-adjoint operators

9.Elements of nonlinear functional analysis

9.1 Fréchet and Gȃteaux derivatives
9.2 Lyapunov–Schmidt method
9.3 Critical point of a functional
9.4 Contracting mappings
9.5 Newton iteration process
9.6 Some applications to PDE

参考教材

Linear Functional Analysis by B. Rynne and M. Youngson, Springer, 2nd edition, 2007.

OTHER REFERENCES

  1. Lax P., Functional Analysis, Wiley-Interscience, 2002
  2. MacCluer, Elementary Functional Analysis, springer, 2009
  3. Lebedev L., Vorovich I., Gladwell G., Functional Analysis (Applications in Me-chanics and Inverse Problems), 2nd edition, Kluwer Academic Publishers, 2002;
  4. Lebedev L., Vorovich I., Functional Analysis in Mechanics, Springer, 2002.